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On the critical dissipative quasi-geostrophic equation. (English) Zbl 0989.86004
The existence and uniqueness of global classical solutions of the critical dissipative quasi-geostrophic (QG) equation \[ \frac{\partial\vartheta}{\partial t} + u \cdot \nabla \vartheta + \kappa (-\Delta)^\alpha \vartheta = 0, \] for initial data that have small \(L^\infty \) norm are proved in the paper. Here \(\alpha \in [ {0,1}]\), \(\kappa > 0\) is the dissipative coefficient, and the 2D velocity field \(u = (u_1 ,u_2)\) is determined from \(\vartheta\) by a stream function \(\psi \) via the auxiliary relations \[ (u_1 ,u_2)= \left(\frac{\partial \psi} {\partial \chi _2},\frac{\partial \psi}{\partial \chi _1}\right),\qquad (-\Delta)^{1/2}\psi = \vartheta. \] QG equation with \(\alpha = \frac{1} {2}\) is the critical dissipative QG equation. Criticality means that the dissipation balances nonlinearity when one takes into account the conservation laws. If no smallness condition is imposed on the initial data, then the issue of global existence for arbitrary data is open. It is shown in the paper that if the \(L^\infty\)-norm of the initial data is small, then the QG equation possesses a global solution in the critical case \(\alpha = \frac{1} {2}\) . The QG equations (dissipative or not) have global weak solutions for arbitrary \(L^2\) initial data. If the initial data is smooth enough, then the solution of the critical dissipative QG equation is unique, smooth, and decays in time. When \(\alpha > \frac{1} {2}\), the smallness assumption on the data is not needed for global existence of smooth solutions for QG equation. This can be proved using the same ideas as for the critical case.

MSC:
86A05 Hydrology, hydrography, oceanography
35Q35 PDEs in connection with fluid mechanics
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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